Algebraic analysis of non-Hermitian quadratic Hamiltonians

TL;DR

This paper analyzes non-Hermitian quadratic Hamiltonians without -symmetry, identifying conditions for real eigenvalues and exceptional points using algebraic methods, and introduces an algebraic alternative to the Bogoliubov transformation.

Contribution

It provides an algebraic framework for analyzing non-Hermitian quadratic Hamiltonians and introduces a new algebraic method as an alternative to the Bogoliubov transformation.

Findings

Conditions for real eigenvalues are derived.

Locations of exceptional points are identified.

An algebraic method simplifies quadratic operators.

Abstract

We study a general one-mode non-Hermitian quadratic Hamiltonian that does not exhibit $\mathcal{PT}$-symmetry. By means of an algebraic method we determine the conditions for the existence of real eigenvalues as well as the location of the exceptional points. We also put forward an algebraic alternative to the generalized Bogoliubov transformation that enables one to convert the quadratic operator into a simpler form in terms of the original creation and annihilation operators. We carry out a similar analysis of a two-mode oscillator that consists of two identical one-mode oscillators coupled by a quadratic term.